NO! NO! Albert R Meyer February 13, 2012 Albert R Meyer February - - PDF document

• Lec 2M.1

Albert R Meyer February 13, 2012

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Mathematics for Computer Science

MIT 6.042J/18.062J

The Well Ordering Principle

.

Lec 2M.2 Albert R Meyer February 13, 2012

Well Ordering principle

Every nonempty set of nonnegative integers has a least element.

Familiar? Now you mention it, Yes. Obvious? Yes. Trivial? Yes. But watch out:

Lec 2M.3 Albert R Meyer February 13, 2012

Well Ordering principle

Every nonempty set of nonnegative integers has a least element. p y integers

rationals

NO!

Lec 2M.4 Albert R Meyer February 13, 2012

Well Ordering principle Every nonempty set of nonnegative integers has a least element.

NO!

Lec 2M.5 Albert R Meyer February 13, 2012

What is the

• youngest age of MIT

graduate?

• smallest # neurons in

any animal?

• smallest #coins = $1.17?

Lec 2M.6 Albert R Meyer February 13, 2012

N ::= nonnegative integers

For rest of this talk, “number” means nonnegative integer

• Lec 2M.7

Albert R Meyer February 13, 2012

2 proof used Well Ordering

m Proof: …suppose 2 = n …can always find such m, n>0 without common factors…

why always ?

Lec 2M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 7

Proof using Well Ordering

Find smallest number m s.t.

m 2 = . If m, n had a n

common factor, c>1, then

( )

m / c 2 =

and m/c m

Albert R Meyer February 13, 2012

( )

<

n / c

Lec 2M.8 Lec 2M M 8 M 8 M 8 M 8 M 8 8 M 8 8 Lec 2M.9 Albert R Meyer February 13, 2012

Proof using Well Ordering

Find smallest number m s.t.

m 2 = . n

This contradiction implies m, n have no common factors.

Lec 2M 9 M 9 M 9 M 9 9 M 9 9 M 9 M 9 9 M 9 M 9 M

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